A Provably-Correct and Robust Convex Model for Smooth Separable NMF

TL;DR

This paper introduces a convex model for smooth separable NMF that guarantees accurate factor recovery even with noisy data, improving robustness and solution uniqueness in practical applications.

Contribution

It proposes a novel convex formulation for smooth SNMF, providing provable correctness and robustness, and adapts a fast gradient method for efficient solution.

Findings

The convex model accurately recovers factors in noisy scenarios.

The method outperforms state-of-the-art algorithms on synthetic data.

Effective on hyperspectral datasets with noise.

Abstract

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.